In iterative image reconstruction methods with regularization or image de-noising methods the problem is commonly formulated as a cost function consisting of a data term and a regularization term.
In statistical iterative reconstruction, the data term is based on a statistical model of the performed measurements while the regularization term incorporates a-priory knowledge about the image to reconstruct. The magnitude of the data term is affected by the data fidelity, which locally depends on the fidelity (variance) of the measurements contributing to the image at a certain location. The a-priori knowledge applied in the regularization typically favors smooth images (low frequency components over high frequencies) to accomplish noise reduction and is weighed by a factor β.
The statistical model for the performed measurements typically results in a certain balance/weighting between the contributions of the measurements and the a-priori model (penalty). The balancing weight considerably varies spatially over the image, resulting in non-uniform resolution or SNR respectively.
However the statistical weights also vary considerably among the measurements contributing to a single image location. The measurements thereby represent line integrals of rays through the voxel under different angles. Due to different weighting, those rays with larger weights contribute more in comparison to the regularization. The regularization penalty typically has an isotropic and spatially invariant characteristic, this in turn leads to anisotropic resolution and noise properties; the resolution perpendicular to those rays with larger weights is higher. This effect can be observed especially at the rim of objects, where rays tangential to the object's border see consistently less attenuation and thus less measurement noise compared to rays perpendicular to the border.
These are a well-known facts, and many researchers have tried to tackle this problem, at which the majority of literature deals with methods that try to achieve a homogeneous and isotropic resolution (cf. the dissertation “Fast Regularization Design for Tomographic Image Reconstruction for Uniform and Isotropic Spatial Resolution” by H. R. Shi, The University of Michigan (2008), and the article “A Penalized-Likelihood Image Reconstruction Method for Emission Tomography, Compared to Post Smoothed Maximum-Likelihood with Matched Spatial Resolution” by J. Nuyts et. al., IEEE Transactions on Medical Imaging, volume 22, pages 1042 to 1052 (2003), the article “Spatial Resolution Properties of Penalized-Likelihood Image Reconstruction: Space-Invariant Tomographs” by J. A. Fessler et. al., IEEE Transactions on Image Processing, volume 5, pages 1346 to 1358 (1996), the article “Regularization for uniform spatial resolution properties in penalized-likelihood image reconstruction” by J. Stayman et. al., Medical Imaging, IEEE Transactions on, volume 19, pages 601 to 615 (2000), the article “Analytical approach to regularization design for isotropic spatial resolution” by J. Fessler, Nuclear Science Symposium Conference Record, 2003 IEEE, volume 3, pages 2022 to 2026 (2003), and the article “Quadratic Regularization Design for 2-D CT” by H. Shi et. al., Medical Imaging, IEEE Transactions on, volume 28, pages 645 to 656 (2009)). All these methods are based on a modification of the regularization term.
Uniformity and isotropy of noise properties have only recently been studied for iterative reconstruction. As for resolution these methods are based on a modification of the regularization term and, more specifically, on linear regularization terms, such as “Quadratic Regularization Design for 3D Axial CT” by Jang Hwan Cho et al., 12th International Conference on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, pp 78-81 aiming at uniform noise characteristics and “Quadratic Regularization Design for 3D Axial CT: Towards Isotropic Noise” by Jang Hwan Cho et al., Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), 2013 IEEE, M22-31 aiming at isotropic noise characteristics.